1. Field of the Invention
The present invention relates in general to the field of information processing, and more specifically to a system and method for quantizing signals using jointly nonlinear delta-sigma modulators.
2. Description of the Related Art
Many signal processing systems include delta sigma modulators to quantize an input signal into one or more bits. Delta sigma modulators trade-off increased noise in the form of quantization error in exchange for high sample rates and noise shaping. “Delta-sigma modulators” are also commonly referred to using other interchangeable terms such as “sigma-delta modulators”, “delta-sigma converters”, “sigma delta converters”, and “noise shapers”.
FIG. 1 depicts a conventional delta sigma modulator 100 that includes a monotonic quantizer 102 for quantizing a digital input signal x(n), where “x(n)” represents the nth input signal sample. The delta sigma modulator 100 also includes an exemplary fourth (4th) order noise shaping loop filter 104 that pushes noise out of the signal band of interest. The output data of each stage 106(1), 106(2), 106(3), and 106(4) of the filter 104 is represented by respective state variables SV1, SV2, SV3, and SV4 of filter 104. The state variables are updated once during each operational period T. The quantizer input signal s(n) is determined from a linear combination of the state variables in accordance with the topology of filter 104. The complete quantizer input signal s(n) is determined from the state variables modified by feed-forward gains d1 through d4 and input signal x(n) modified by feedback coefficient c0 and feed-forward gain d0. For audio signals, the signal band of interest is approximately 0 Hz to 20 kHz. The four feedback coefficients c0, c1, c2, and c3 and/or the feed-forward coefficients set the poles of both the noise transfer function (NTF) and the signal transfer function (STF) of filter 104. In general, there are two common filter topologies, feed-forward and feedback. In the feedback case, the feed-forward coefficients are all zero, except for the last coefficient. In the feed-forward case, all of the feedback coefficients are 0, except for c0, which is usually defined as 1, without any loss of generality. The NTF of filter 104 has four (4) zeros at DC (0 Hz). Local resonators are often added, with feedback around pairs of integrators, in order to move some of the zeroes to frequencies higher in the signal passband. Typical high performance delta sigma modulators include fourth (4th) order and higher loop filters although filter 104 can be any order. The NTF often distributes zeros across the signal band of interest to improve the noise performance of the delta sigma modulator 100.
The topology of each stage is a matter of design choice. Stages 106(i) are each represented by the z-domain transfer function of z−1/(1−z−1). Group 108 is functionally identical to group 110. Stage 106(1) can be represented by a leading edge triggered delay 112 and feedback 114.
FIG. 2 depicts the quantizer 102 modeled as a gain, g, multiplying the quantizer input signal s(n) plus additive white noise n. The quantizer output noise is then modeled as n/(1+z−1*g*H(z)). However, the quantizer output noise model often breaks down because the gain g is actually dependent upon the level (magnitude) of the input signal x(n). Additionally, the additive noise is correlated to the input signal. For low level input signals x(n), a tendency exists for the feedback signal into the quantizer 102 to be low, effectively making the gain high or breaking down the quantizer output noise model altogether. Because one-bit quantizers have no well-defined gain, a high gain for low level quantizer input signals is particularly bad because it can decrease the signal-to-noise ratio (SNR) of the delta sigma modulator 100. Often white noise, or dither, is added to the input of the quantizer in order to aid this situation; however that noise decreases the dynamic range and maximum signal input of the system.
Referring to FIGS. 1, 2, and 3, the quantizer 102 quantizes an input signal x(n) monotonically by making a decision to select the closest feedback value to approximate the input signal. In a one-bit delta sigma modulator, the quantizer has only two legal outputs, referred to as −1 and +1. Therefore, in a one-bit embodiment, quantizer 102 quantizes all positive input signals as a +1 and quantizes all negative input signals as −1. The quantization level changeover threshold 304 is set at DC, i.e. 0 Hz, and may be quantized as +1 or −1.
FIG. 3 graphically depicts a monotonic, two-level quantization transfer function 300, which represents the possible selections of each quantizer output signal y(n) from each quantizer input signal s(n). The diagonal line 302 depicts a monotonic unity gain function and represents the lowest noise quantization transfer function. “Monotonic” is defined by a function that, as signal levels increase, consists of either increasing quantizer output state transitions (“transitions”) or decreasing transitions, but not both increasing and decreasing transitions. To mathematically define “monotonically increasing” in terms of quantization, if the transfer function of the quantizer 102 is denoted as Q(s), then Q(s1)≧Q(s2), for all s1>s2, where “s1” and “s2” represent quantizer input signals. Mathematically defining “monotonically decreasing” in terms of quantization, if the transfer function of the quantizer 102 is denoted as Q(s), then Q(s1)≧Q(s2), for all s1<s2. Thus, in general, a monotonic quantization transfer function must adhere to Equation 1:Q(s1)>Q(s2), for all |s1|>|s2|.  [Equation 1]
In many cases, dithering technology intentionally adds noise to the quantizer input signal s(n) to dither the output decision of quantizer 102. Adding dithering noise can help reduce the production of tones in the output signal y(n) at the cost of adding some additional noise to the delta sigma modulator loop because the quantization noise is generally increased. However, adding dithering noise to the quantizer does not convert a monotonic quantization transfer function into a non-monotonic quantization transfer function. Adding dithering noise merely changes the probability of some quantizer decisions. An alternative perspective regarding dither is to simply add a signal prior to quantization, which has no effect on the quantization transfer function.
Magrath and Sandler in A Sigma-Delta Modulator Topology with High Linearity, 1997 IEEE International Symposium on Circuits and Systems, Jun. 9–12, 1987 Hong Kong, (referred to as “Magrath and Sandler”) describes a sigma-delta modulator function that achieves high linearity by modifying the transfer function of the quantizer loop to include bit-flipping for small signal inputs to the quantizer. Magrath and Sandler discusses the compromise of linearity of the sigma-delta modulation process by the occurrence of idle tones, which are strongly related to repeating patterns at the modulator output and associated limit cycles in the system state-space. Magrath and Sandler indicates that injection of a dither source before the quantizer is a common approach to linearise the modulator. Magrath and Sandler discusses a technique to emulate dither by approximately mapping the dither onto an equivalent bit-flipping operation.
FIG. 4 graphically depicts the single non-monotonic region quantization transfer function 400 that emulates dither as described by Magrath and Sandler. Quantizer function 400 is necessarily centered around s(n)=0, as described by Magrath and Sandler, to emulate conventional dither. According to Magrath and Sandier, if the absolute value of the input (“|s(n)|” in FIG. 1) to the quantizer is less than B, a system constant, then the quantizer state is inverted as depicted by quantizer function 400.
Input signals s(n) to the quantizer 102 can be represented by probability density functions (PDFs). FIG. 5A depicts PDFs of each quantizer input signal s(n) during operation at small and large input signal levels. PDF 502 represents small signal levels for each signal s(n). The narrow PDF 502 can indicate high delta sigma modulator loop gain g. As the magnitude of signal levels for signal s(n) increase, the PDF of each signal s(n) changes from the narrow PDF 502 to the wider PDF 504.
FIG. 5B depicts a near ideal PDF 500 for each quantizer input signal s(n) because all signals are clustered around the quantization levels +1 and −1. Accordingly, the quantization noise n (error) is very small.
FIG. 6 graphically depicts a convex region 602 and a nonconvex region 604. A set in Euclidean space is a convex set if the set contains all the line segments connecting any pair of points in the set. If the set does not contain all the line segments connecting any pair of data points in the set, then the set is nonconvex (all referred to as “concave”). The convex region 602 includes all sets of data points within the boundaries of convex region 602. As depicted by the example line segments AB and CD, all line segments in convex region 602 connecting any pair of data points are completely contained within convex region 602
Region 604 represents a nonconvex region because there exists at least one line segment AB connecting a pair points {A,B} that is not completely contained within region 603. Thus, by definition, region 604 is a nonconvex region.
FIG. 7 depicts the interrelationship of two state variables SVx and SVy with respect to the output y(n) of a monotonic quantizer 102 (FIG. 1). State variables SVx and SVy represent any respective state variable of filter 104. Line 702 represents the boundary between the +1 quantization region and the −1 quantization region for a quantization output level. The boundary 702 of the quantization regions +1 and −1 is characterized by a linear interrelationship between state variables SVx and SVy. The quantization regions +1 and −1 are also defined by convex boundaries. In the case of a feedback topology for the loop filter, the quantizer is responsive to only one state variable, that being the last (or highest order) one. In the case of a feed-forward filter topology, the relationship illustrated in FIG. 7 is active. In general, the regions will have dimensionality of the order of the filter, e.g., a fourth order filter would have a 4-space diagram. The 2 dimensional diagrams are meant to be representative of a 2-dimensional slice of an actual n-dimensional region functions being depicted.
FIG. 8 depicts the interrelationship between two state variables SVx and SVy with respect to the output y(n) of a bit-flipping quantizer 102. The boundaries between quantization regions A, B, C, and D are linear. The non-monotonic, bit-flipping quantizer 102 has four, convex quantization regions that alternate twice between +1 and −1. The four quantization regions are also defined by a linear relationship for any pair of state variables SVx and SVy. Again, in a higher order system, the actual regions depicted are n-dimensional.